Ring of integers

In mathematics, the ring of integers of an algebraic number field K is the ring of all integral elements contained in K. An integral element is a root of a monic polynomial with rational integer coefficients, xⁿ + c_n−1xⁿ⁻¹ + … + c₀ . This ring is often denoted by O_K or ${\displaystyle {\mathcal {O}}_{K}}$. Since any rational integer number belongs to K and is its integral element, the ring Z is always a subring of O_K.

The ring Z is the simplest possible ring of integers. Namely, Z = O_Q where Q is the field of rational numbers. And indeed, in algebraic number theory the elements of Z are often called the "rational integers" because of this.

...
Wikipedia